Analysis on curve negotiating ability of rail grinder in grinding state

Rail grinding becomes an important maintenance means of railway. Dynamic behavior of rail grinder is due to vehicle-track coupling relationship based on mechanical-electric-hydraulic coupling. Curve negotiating ability of rail grinder through modeling and simulation based on one-side grinding is studied in this paper. Simulation result is shown below. In typical case, rail grinding will increase transverse displacement of wheelsets, derailment coefficient of wheels in front wheelset, and unloading rate of wheelsets. In other case, increase of rail irregularity amplitude and decrease of its wavelength, which aggravates fluctuation of grinding power, has little influence on curve negotiating ability. When line radius curvature decreases, compared to state without grinding, decline to curve negotiating ability of state with grinding is more significantly. When number of grinding wheels at work increases, lateral displacement of wheelset, derailment coefficient of wheels in front wheelset, and unloading ratio of wheelsets increase. In short, rail grinding will significantly deteriorate curve negotiating ability of rail grinder.

www.nature.com/scientificreports/ affect vibration state of buggy and grinding car Dynamic behavior of buggy and grinding car will affect grinding pressure also. In order to ensure stability of grinding, grinding process needs to be controlled. Constant power grinding is generally adopted, which is accomplished by cooperation of hydraulic system and control system. Hydraulic system is equipped with pilot proportional reducing valve. Under action of it, pressure of hydraulic system is directly proportional to control voltage, but there is a lag. Control system obtains actual grinding power by detecting current of grinding motor. Pressure of hydraulic system is controlled by adjusting control voltage of that valve in real time through closed-loop feedback. When grinding power is low, system will keep pressure of rodless cavity greater than that of rod cavity, pushing rod of hydraulic cylinder out. Extension of hydraulic cylinder will produce greater Hertzian contact force and increase grinding power. And the opposite is the same. But stability control to grinding power is a dynamic process, not absolutely constant.

Model of rail grinder
Submodel of vehicle-track dynamic coupling. Under grinding mode, grinding car and buggy are connected through draw bar, and suspension hydraulic cylinder share no load at this time. Grinding wheel is in contact with rail, and the whole vehicle runs at a uniform speed. Different rail grinders have different structures, but they are all composed of grinding car and buggy, both of which confirm to Zhai-Sun model, that is vehicle track dynamic coupling model. There are many direction and pressure adjustment hydraulic cylinders in buggy. When hydraulic oil is charged, hydraulic cylinder can be equivalent to spring damping structure. Draw bar can also be equivalent to spring damping structure. The whole vehicle track coupling model of rail grinder is shown in Fig. 2, while point O is the rotation center of cradle.
Equivalent stiffness 20 of charged hydraulic cylinder is: Displacement of the two rigid bodies on both sides of draw bar are:   Torque at direction adjusting hydraulic cylinder is: Force at grinding hydraulic cylinder is: When calculating contact force between grinding wheel and rail, grinding wheel can be regarded as a plane and rail can be regarded as a curved surface. According to Carter theory 21 , width of contact band can be shown in Eq. (8) and contact pressure can be shown in Eq. (9).
Force between rail and grinding wheel is: For force of other parts, please refer to reference 8 . Motion equations of all parts can be obtained through Newton's or Dalamber's theorem. At this time, rail grinder includes 20 rigid bodies: 1 grinding car, 1 buggy frame, 2 bogies, 4 wheelsets, 4 cradles and 8 grinding wheels. Cradle only has rotation DOF, and grinding wheel only has extension DOF. For other parts, 5 DOF are considered: transverse displacement, vertical displacement, rolling, pitching and yawing. So the whole rail grinder has 52 DOF. Submodel of hydraulic system. Figure 3 shows the hydraulic model. Oil volume will affect oil pressure, and their relationship is shown in Eq. (11). Equations below can be inferred from Eq. (11). Equation (12) is the force balance of pilot valve. Equation (13) is the calculation method of electromagnetic force. Equation (14) is the formula for calculating pilot valve pressure through hydraulic flow. Equation (15) is the force balance formula of main valve. Equation (16) is the flow formula of main valve when oil is injected to hydraulic cylinder. Figure 3. Hydraulic model.  Figure 4 shows the control model. www.nature.com/scientificreports/ Rail grinder runs on rail with uniform speed, and coordinates of upper hinge joint on grinding hydraulic cylinder are exported through vehicle-track model. Control model exports control voltage u to hydraulic model by detecting grinding motor current I. Hydraulic model receives control voltage u and adjusts pressure of hydraulic system. As a result, extension L of hydraulic cylinder changes under influence of pressure, and it is exported also. After receiving coordinates and extension amount L, program calculates compression amount δ by comparing with rail irregularity. Grinding force can be calculated from compression amount δ. Under action of grinding force, lateral force F y , vertical force F z , lateral torque T y , vertical torque T z can be output to vehicle-track model, and Hertzian force F h can be output to hydraulic system. In this iterative cycle, curve negotiating ability of rail grinder in the whole line can be obtained.

System simulation
Rail grinder involves three parts: vehicle-track system, hydraulic system and control system, which need to be modeled separately. Through interface program, the three can transfer parameters and realize interaction. Figure 5 shows the mechanical-electric-hydraulic coupling model.
Taking Tianjin Metro Line 1 as an example, rail corrugation on upper side rail of curve section is serious. It is mainly composed of long wave, and the amplitude can reach mm level 23 . The line selected in this paper is shown in Table 2, and standard sinusoidal excitation is applied as vertical irregularity of rail to imitate rail corrugation, as shown in Eq. (24). In this paper, we simulate grinding mode of rail top surface trimmed, that is, set angle of cradle is 0°. According to actual rail damage, method of outer rail one-side grinding is adopted. Curve negotiating ability can be analyzed by simulation.
During rail grinding, change of grinding parameters will affect curve negotiating ability of rail grinder. In order to evaluate influence of those parameters, 4 cases are set. At this moment, we focus on the peak value. Table 3 shows grinding parameters and cases. According to function of rail grinder, evaluation and test method of its dynamic behavior shall comply with GB/T 17426-1998. All dynamic behavior talking inside this paper are based on this standard. Unspecified dynamic characteristics in this paper refer to that of buggy. "with grinding" refers to dynamic characteristics of buggy in grinding state, and "without grinding" refers to dynamic characteristics of buggy not in grinding state.

Analysis of grinding power and curve negotiating ability
Typical case. Figures 6,7,8,9, 10 and 11 shows mechanical-electric-hydraulic coupling relationship of rail grinder under typical case. Phase difference for grinding wheels at different initial positions are different, and fluctuation amplitude of grinding power are also different. Phase difference is caused by non-coincidence of grinding wheel and buggy wheel, and fluctuation amplitude of grinding power is caused by pitching motion of buggy frame. For constant power grinding, phase difference and power fluctuation are unfavorable factors.
There is hydraulic pressure in grinding hydraulic cylinder and Hertz force in the grinding wheel. This structure can be approximately regarded as a series spring. Length of this spring is the value that vertical coordinate Z o of cradle minus extension length L of hydraulic cylinder and rail irregularity. Grinding power is proportional www.nature.com/scientificreports/ to difference between spring length and spring nominal length. If grinding power is regarded as output, Z o and rail irregularity can be regarded as passive input, while L can be regarded as positive input. So grinding power can be controlled by L. Pressure of rodless cavity in grinding hydraulic cylinder is directly proportional to difference between its accumulated flow Q and A a L . Pressure of rod cavity is directly proportional to sum between its accumulated flow Q and A b L . If L is regarded as passive input temporarily, Q of rodless cavity can be regarded as positive input, pressure of rodless cavity can be regarded as positive output, Q of rod cavity can be regarded as passive input, and pressure of rod cavity can be regarded as passive output. Output L is controlled by pressure difference between the two cavities. Control voltage is proportional to pressure of rodless cavity, which is realized by changing flow Q of rodless cavity. As can be seen from Fig. 8, control voltage is reversed with pressure of rodless cavity to achieve grinding power fluctuation control.
Control model Hydraulic model Grinding force Vehicle model   www.nature.com/scientificreports/ For rail surface grinding, grinding power is directly proportional to vertical force F z and torque T z from rail to grinding wheel. Transverse force F y and torque T y from rail to grinding wheel can be ignored. Curve negotiating ability can be obtained by inputting F z and T z into the vehicle track model. Figures 12, 13, 14, 15, 16 and 17 shows curve negotiating ability of buggy under typical case. Because wheelbase of buggy is large, in curve line, transverse displacement of front wheelset is negative and that of back wheelset is positive. It can be seen from Fig. 12, in curve line transverse displacements of both wheelsets increase, but it is not obvious. This is caused by vertical grinding force exerted on one-side way and grinding torque in z direction. And, torque in x direction from on-side load produced more contributes.
When work, speed of rail grinder is below 20 km/h, while that of metro vehicle can reach 80 km/h. In existing line, to rail grinder, superelevation is always surplus, so transverse force is positive. Due to existence of vertical grinding pressure, it can share and reduce wheel rail vertical force. Considering one-side load of grinding vertical force, load reduction effect of left wheel is greater than that of right wheel. From Figs. 13 and 14, for left wheel of    As it can be seen from Fig. 17, for front wheelset, when enter transition section, vertical force of left wheel increases and that of right wheel decreases. When leave transition section, it is just opposite. And the opposite is true for back wheelset. When influence of vertical grinding pressure from one-side load is exerted, compared to state without grinding, vertical force of left wheel reduces, and that of right wheel is basically unchanged. For the whole line, maximum value of unloading ratio increases. That of front wheelset increases by about 40.2%, and back wheelset increases by 34.2%, compared to state without grinding.        www.nature.com/scientificreports/ Influence of grinding parameters. For studying influence of different grinding parameters, this paper extracts peak points to analyze. Figures 18, 19, 20 and 21 shows curve negotiating ability of buggy in case 1. As irregularity amplitude increases, fluctuation of grinding power increases. There is little change in wheelset lateral displacement, derailment coefficient and unloading ratio. Increase of rail irregularity increases fluctuation of Hertzian force between grinding wheel and rail, so fluctuation amplitude of grinding power increases. Uneven grinding will worsen grinding effect, which is an adverse effect. Within the given amplitude range of irregularity, fluctuation of Hertzian force is not enough to affect curve negotiating ability of buggy. Under the influence of primary longitudinal stiffness, lateral displacement of front wheelset is obviously greater than that of back wheelset. Since buggy has the following two characteristics:1.Compared with ordinary train, travelling speed is slow. When pass curve section, the superelevation of line is always too high, which is disadvantageous to curve negotiation ability. 2. 8 grinding wheels need to be installed on buggy. Wheelbase of buggy is generally large to reserve space for grinding wheels, which is also a disadvantageous factor. The above characteristics further increase the difference in lateral displacement between front and back wheelset. Figures 22, 23, 24 and 25 shows curve negotiating ability of buggy in case 2. As irregularity wavelength reduces, fluctuation of grinding power increases obviously. There is little change in wheelset lateral displacement. Derailment coefficient generally changes little, and that of left wheel in front wheelset increases slightly. Unloading ratio increases, but it is not obvious. Decrease of the wavelength of rail irregularity, increases fluctuation frequency of Hertzian force between grinding wheel and rail, and increases vibration amplitude of buggy. Finally, fluctuation amplitude of grinding power increases significantly and grinding effect is deteriorated. Within the     www.nature.com/scientificreports/ given wavelength range of irregularity, fluctuation of Hertzian force is not enough to affect curve negotiating ability of buggy. Figures 26, 27, 28 and 29 shows curve negotiating ability of buggy in case 3. As curvature radius decreases, fluctuation power decreases slightly. Transverse displacement of wheelset increases, and that of back wheelset is very obvious. Derailment coefficient increases, and that of left wheel in front wheelset increases very obviously. Unloading ratio is differentiated. Without grinding, unloading ratio of wheelset increases with decrease of curvature radius. With grinding, that of front wheelset decreases and back wheelset increases when curvature radius decreases. This is mainly due to influence of one-side grinding that vertical force exerted on transition sections. Dynamic behavior when enter transition section and leave are different. With decrease of curve curvature radius, curve negotiating ability of states with grinding and without grinding both decrease, but decline of the state with grinding is greater. Grinding process deteriorates curve negotiating ability. Figures 30, 31, 32 and 33 shows curve negotiating ability of buggy in case 4. As number of grinding wheels at work increases, fluctuation of grinding power has little changes which is only related to grinding wheel position. It generally conforms to behavior of independent grinding. While number of grinding wheels at work increases, lateral displacement of wheelset increases, and increase range of front wheelset is more obvious than that of back wheelset. Derailment coefficient increases, and that of left wheel in front wheelset is more obvious than that of right wheel in front wheelset. Unloading ratio also increases. Increasing number of grinding wheels at work will significantly increase total grinding pressure, which will inevitably affect curve negotiating ability of rail grinder.       www.nature.com/scientificreports/

Conclusion
Dynamic behavior of rail grinder is due to vehicle-track coupling relationship built on mechanical-electrichydraulic coupling. Modeling and simulation of the whole system under outer rail one-side grinding are exerted in this paper. The result is shown as below.
In typical case, transverse displacement of wheelsets in buggy increases. To left wheel of front wheelset, lateral force increases and vertical force decreases, so derailment coefficient increases. To right wheel of front wheelset, lateral force increases and vertical force changes little, and derailment coefficient still increases. Unloading ratio of wheelsets increases. This is mainly caused by vertical grinding force exerted on one-side way making wheeltrack contact relationship changed.
In other 4 cases, irregularity amplitude and wavelength only affect fluctuation of grinding power. Increase of irregularity amplitude and decrease of irregularity wavelength will aggravate fluctuation of grinding power, but they have little effect on curve negotiating ability of rail grinder. Reduction of curvature radius has little effect on fluctuation of grinding power, but it increases transverse displacement of wheelsets and derailment coefficient of wheels in front wheelset, reduces unloading rate of front wheelset and increases unloading rate of back wheelset. Curve negotiating ability under the state with grinding and without grinding are all decreased but declined range of state with grinding is greater. Increase of grinding wheels at work will increase lateral displacement of wheelsets, derailment coefficient of wheels in the front wheelsets, and unloading rate of wheelsets.
In short, rail grinding has an impact on dynamic behavior of rail grinder, which will significantly deteriorate its curve negotiating ability. And curve negotiating ability in grinding state is still inside the range of standard.

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.